SOLUTION: cos(x+(π/2))-cos(2x)=0

Algebra ->  Algebra  -> Trigonometry-basics -> SOLUTION: cos(x+(π/2))-cos(2x)=0      Log On

Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!
Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!

   

Question 429870: cos(x+(π/2))-cos(2x)=0
Found 2 solutions by robertb, Gogonati:
Answer by robertb(4012) About Me  (Show Source):
You can put this solution on YOUR website!
cos%28x%2B+pi%2F2%29+=+cos%282x%29
In the interval [0, 2pi), either
x%2B+pi%2F2+=+2x, or x+%2B+pi%2F2+=+2%2Api+-+2x
==> x+=+pi%2F2, or
3x+=+%283%2Api%29%2F2, or x+=+pi%2F2 (Same as the first solution.)
Therefore the general solution is pi%2F2+%2B+2n%2Api, where n is any integer.

Answer by Gogonati(817) About Me  (Show Source):
You can put this solution on YOUR website!
Solution:From the trigonometry we know that: cos2x=%28cos%28x%29%29%5E2-%28sin%28x%29%29%5E2
%28cos%28x%29%29%5E2=1-%28sin%28x%29%29%5E2, and cos(90+x)=-sin(x). Now we write:

2%28sin%28x%29%29%5E2-sin%28x%29-1=0, substitute six=a and rewrite the equation:
2a%5E2-a-1=0, which is a quadratic equation. Solve this equation:
There are two roots: a=1 and a=-1/2, thus we have: sin(x)=1 and sin(x)=-12
Solve now these two trigonometric equations and find:
x=2n*pi+ pi/2,
x=2n*pi-pi/6, and x=2n*pi+7pi/6
Done.